Calculates the local/moving non-parametric regression (i.e., LOESS, LOWESS, etc.) forecast.
Syntax
NxLOCREG(X, Y, P, Kernel, Alpha, Optimize, Target, Return)
- X
- is the x-component of the input data table (a one-dimensional array of cells (e.g., rows or columns)).
- Y
- is the y-component (i.e., function) of the input data table (a one-dimensional array of cells (e.g., rows or columns)).
- P
- is the polynomial order (0 = constant, 1 = linear, 2 = Quadratic, 3 = Cubic, etc.), etc.). If missing, P = 0.
- Kernel
- is the weighting kernel function used with KNN-Regression method : 0(or missing) = Uniform, 1 = Triangular, 2 = Epanechnikov, 3 = Quartic, 4 = Triweight, 5 = Tricube, 6 = Gaussian, 7 = Cosine, 8 = Logistic, 9 = Sigmoid, 10 = Silverman.
Value Kernel 0 Uniform Kernel (default). 1 Triangular Kernel. 2 Epanechnikov Kernel. 3 Quartic Kernel. 4 Triweight Kernel. 5 Tricube Kernel. 6 Gaussian Kernel. 7 Cosine Kernel. 8 Logistic Kernel. 9 Sigmoid Kernel. 10 Silverman Kernel. - Alpha
- is the fraction of the total number (n) of data points that are used in each local fit (between 0 and 1). If missing or omitted, Alpha = 0.333.
- Optimize
- is a flag (True/False) for searching and using optimal bandwidth (i.e., fraction value or $\alpha$). If missing or omitted, optimize is assumed to be False.
- target
- is the desired x-value(s) to interpolate for (a single value or a one-dimensional array of cells (e.g., rows or columns)).
- Return
- is a number that determines the type of return value: 0 = Forecast (default), 1 = errors, 2 = Smoothing parameter (bandwidth), 3 = RMSE (CV). If missing or omitted, NxREG returns forecast/regression value(s).
Return Description 0 Forecast/Regression value(s) (default). 1 Forecast/Regression error(s). 2 Kernel Smoothing parameter (bandwidth). 3 RMSE (cross-validation).
Remarks
- Local regression is a non-parametric method combining multiple regression models in a k-nearest-neighbor-based meta-model.
- Local regression or local polynomial regression, also known as moving regression, is a generalization of moving average and polynomial regression.
- Its most common methods initially developed for scatterplot smoothing are LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing).
- Outside econometrics, LOESS is known and commonly referred to as Savitzky–Golay filter. Savitzky–Golay filter was proposed 15 years before LOESS.
- $\alpha$ is called the smoothing parameter because it controls the flexibility of the LOESS regression function. Large values of ${\displaystyle \alpha }$ produce the smoothest functions that wiggle the least in response to fluctuations in the data. The smaller ${\displaystyle \alpha }$ is, the closer the regression function will conform to the data. Using too small a value of the smoothing parameter is not desirable, however, since the regression function will eventually start to capture the random error in the data.
- The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variable (X).
- Observations (i.e., rows) with missing values in X or Y are removed.
- NxLOCREG is related to NxKREG, but NxLOCREG uses the nearest K-NN points to calculate kernel bandwidth and conduct its local regression.
- The time series may include missing values (e.g., #N/A) at either end.
- The NxLOCREG() function is available starting with version 1.66 PARSON.
Files Examples
Related Links
References
- Pagan, A.; Ullah, A. (1999). Nonparametric Econometrics. Cambridge University Press. ISBN 0-521-35564-8.
- Simonoff, Jeffrey S. (1996). Smoothing Methods in Statistics. Springer. ISBN 0-387-94716-7.
- Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. ISBN 0-691-12161-3.
- Henderson, Daniel J.; Parmeter, Christopher F. (2015). Applied Nonparametric Econometrics. Cambridge University Press. ISBN 978-1-107-01025-3.
- Gyorfi, L., Kohler, M., Krzyzak, A. & Walk, H. (2002), A distribution-free Theory of Nonparametric Regression, Springer, New York.
- Hastie, T. & Tibshirani, R. (1990), Generalized Additive Models, Chapman and Hall, London.
- Hastie, T., Tibshirani, R. & Friedman, J. (2009), The Elements of Statistical Learning; Data Mining, Inference, and Prediction, Springer, New York. Second edition.
- Johnstone, I. (2011), Gaussian estimation: Sequence and wavelet models, Under contract to Cambridge University Press. Online version at http://www-stat.stanford.edu/~imj.
- Kim, S.-J., Koh, K., Boyd, S. & Gorinevsky, D. (2009), ‘`1 trend filtering’, SIAM Review 51(2), 339–360.
- Kimeldorf, G. & Wahba, G. (1970), ‘A correspondence between Bayesian estimation on stochastic processes and smoothing by splines’, Annals of Mathematical Statistics 41(2), 495–502.
- Lin, Y. & Zhang, H. H. (2006), ‘Component selection and smoothing in multivariate nonparametric regression’, Annals of Statistics 34(5), 2272–2297.
- Mallat, S. (2008), A wavelet tour of signal processing, Academic Press, San Diego. Third edition.
- Mammen, E. & van de Geer, S. (1997), ‘Locally adaptive regression splines’, Annals of Statistics 25(1), 387–413.
- Raskutti, G., Wainwright, M. & Yu, B. (2012), ‘Minimax-optimal rates for sparse additive models over kernel classes via convex programming, Journal of Machine Learning Research 13, 389–427.
- Ravikumar, P., Liu, H., Lafferty, J. & Wasserman, L. (2009), ‘Sparse additive models’, Journal of the Royal Statistical Society: Series B 75(1), 1009–1030.
- Scholkopf, B. & Smola, A. (2002), ‘Learning with kernels’.
- Silverman, B. (1984), ‘Spline smoothing: the equivalent variable kernel method’, 12(3), 898–916.
- Simonoff, J. (1996), Smoothing Methods in Statistics, Springer, New York.
- Steidl, G., Didas, S. & Neumann, J. (2006), ‘Splines in higher-order TV regularization’, International Journal of Computer Vision 70(3), 214–255.
- Stone, C. (1985), ‘Additive regression models and other nonparametric models’, Annals of Statistics 13(2), 689–705.
- Tibshirani, R. J. (2014), ‘Adaptive piecewise polynomial estimation via trend filtering’, Annals of Statistics 42(1), 285–323.
- Tsybakov, A. (2009), Introduction to Nonparametric Estimation, Springer, New York.
- Wahba, G. (1990), Spline Models for Observational Data, Society for Industrial and Applied Mathematics, Philadelphia.
- Wang, Y., Smola, A. & Tibshirani, R. J. (2014), ‘The falling factorial basis and its statistical properties, International Conference on Machine Learning 31.
- Wasserman, L. (2006), All of Nonparametric Statistics, Springer, New York.
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